It is assumed that a battery of n tests has been resolved into components in a common factor space of r dimensions and a unique factor space of at most n dimensions, where r is much less than n. Simplified formulas for ordinary multiple and partial correlation of tests are derived directly in terms of the components. The best (in the sense of least squares) linear regression equations for predicting factor scores from test scores are derived also in terms of the components. Spearman's “single factor” prediction formulas emerge as special cases. The last part of the paper shows how the communality is an upper bound for multiple correlation. A necessary and sufficient condition is established for the square of the multiple correlation coefficient of test j on the remaining n−1 tests to approach the communality of test j as a limit asn increases indefinitely while r remains constant. Limits are established for partial correlation and regression coefficients and for the prediction of factor scores.

A modification of the method of successive intervals is presented which yields scale values correlating .995 with those from Thurstone's method described by Saffir. Values yielded by the present method can be obtained in 25 per cent of the time required by the older method and are shown to be, on a priori grounds, more reliable as well.

A new approximation method for obtaining a maximized multiple criterion, based on the formula of Edgerton and Kolbe (2), is presented. By applications to examples from the literature (1, 2), the new method is evaluated in comparison with the Horst approximation (1), a suggested revision of the Horst procedure, and the more exact but more laborious iterative method for the principal axis solution of Hotelling.

The study of William Brown and William Stephenson, “A Test of the Theory of Two Factors,” is re-analyzed by means of the Thurstone multiple factor methods. No tests or correlations are left out of the original table of correlations as is done in the original analysis in an attempt to validate the two-factor theory. Space, verbal, and perceptual speed factors similar to those found by Thurstone, Wright, and Garrett are identified. A common factor of “Maturation” is postulated to account for the remaining communality of the tests. A fifth factor is considered to have no significance due to the small amount of variance which it contributes to the total.

In tetrachoric correlation there is a four-fold distribution based on the four combinations of A’s and not —A’s and B’s and not —B’s. The analogous problem in test construction is that of determining the relationship between two items or test questions, upon which all individuals are scored pass or fail.

The fundamental factor theorem is developed in matrix form for the case of correlated factors. The properties of the correlated factor system are discussed, and some effects of sampling error considered. The psychological meaning of correlated factors is discussed, and several mechanisms by which general factors may operate in the factorial system are indicated.

A battery of thirty-six tests was given to a group of high-school seniors. The factorial analysis reveals essentially the same primary factors that were found in previous studies. The test battery reveals a simple structure.

An n-rowed correlation matrix may be thought of as an ellipsoid in n-dimensional space with its center at the origin. The principal components of the matrix are essentially the semi-axes of the ellipsoid. A direct and simple method of computing the lengths and directions of these semi-axes is presented.